Spherical gradient texture on plane not working
Connect Texture > Gradient Texture (spherical) to Diffuse color node, as a vector choose Object and add Color Ramp to play with colors and color blends.



Edit:

For transparency you can choose Color Ramp as factor for Mix Shader.



Edit 2:

I have created scene with the same unit settings as yours and here is what we have:

Plane, scaled to 2x2m.



And this very same setup but with applied scale (CTRL + A).



Edit 3:

I have checked it in Blender 2.77a and 2.78 RC1 and it looks all the same. So my guess (as Blender docs doesn't cover much about Texture Coordinates > Object) it's not a bug but this is how those calculations works. Other textures behave the same. You can 'fix' texture scaling in Texture tab without setting Mapping node and calculations there are correct - scale 100.



Edit 4:

Good catch with other dimensions. Now I'm totally stuned about it. 
But here is a method that is working with every dimension scale (applied or not). Don't ask me why it is working, but it is working :)

You need to change Texture Coordinates to Generated then in Texture tab change Location to X > 5mm and Y > 5mm (this is part that I don't know why 5mm is ok for 2, 20 and 200m, this was my lucky guess) and Scale to 0.5.